Due to the continuity equation and after a monotone class argument, we have the crude estimate
\(\left| \int _{\Gamma } F \mathrm {d}(\mathsf {P}_t-\mathsf {P}_s) \right| \le n \int _s^t \left[\int _{\Gamma \times \mathcal {T}} (|F(\nu +\genfrac{}{}{}2{1}{n}\delta _x)|+|F(\nu )|) \, \mathrm {d}\mathsf {J}_r^{+} + \int _{\Gamma \times \mathcal {T}} (|F(\nu -\genfrac{}{}{}2{1}{n}\delta _x)|+|F(\nu )|) \, \mathrm {d}\mathsf {J}_r^{-} \right] \mathrm {d}r,\) 
for any \(F\in \mathcal {B}_c(\Gamma )\) .
Now fix \(F\in \mathcal {B}_c(\Gamma )\) , and let \(K:=\sup _{\nu \in \Gamma } F(\nu )(1+\nu (\mathcal {T})^2)\) . Note that by the bounds of Lemma REF  for any \(\nu \in \Gamma _n\) , we have the following estimates
\(|F|(\nu )&\le K (1+\nu (\mathcal {T})^2)^{-1}\\|F|(\nu +\genfrac{}{}{}2{1}{n}\delta _x)&\le K \big (1+(\genfrac{}{}{}2{1}{n}+\nu (\mathcal {T}))^2\big )^{-1}\le K (1+\nu (\mathcal {T})^2)^{-1},\\|F|(\nu -\genfrac{}{}{}2{1}{n}\delta _x)&\le K \big (1+(\genfrac{}{}{}2{-1}{n}+\nu (\mathcal {T}))^2\big )^{-1}\le K \max \lbrace 1+\genfrac{}{}{}2{2}{n^2},2\rbrace  (1+\nu (\mathcal {T})^2)^{-1},\) 

and therefore
\(\left| \int _{\Gamma } F \mathrm {d}(\mathsf {P}_t-\mathsf {P}_s) \right| \le 4 n K \max \Bigl \lbrace 1+\genfrac{}{}{}2{2}{n^2},2\Bigr \rbrace  \int _s^t \int _{\Gamma \times \mathcal {T}} (1+\nu (\mathcal {T})^2)^{-1} \mathrm {d}( \mathsf {J}^+_r+\mathsf {J}^-_r) \, \mathrm {d}r.\) 
Taking the supremum over all \(F\in \mathcal {B}_c(\Gamma )\)  with \(\sup _{\nu \in \Gamma } F(\nu )(1+\nu (\mathcal {T})^2)\le 1\)  we conclude that
\(d_{TV,w}(\mathsf {P}_s,\mathsf {P}_t)&=\int _{\Gamma } (1+\nu (\mathcal {T})^2)^{-1}\, \mathrm {d}|\mathsf {P}_t^1-\mathsf {P}_s^2|\\&\le 4 n \max \Bigl \lbrace 1+\genfrac{}{}{}2{2}{n^2},2\Bigr \rbrace  \int _s^t (1+\nu (\mathcal {T})^2)^{-1} (\mathrm {d}\mathsf {J}^+_r+\mathsf {J}^-_r) \, \mathrm {d}r.\) 
Next, suppose that \(\mathsf {P}_t\ll \Pi _n, \mathsf {J}^{\pm }_t\ll \vartheta _{\Pi _n}^{\pm }\)  for all \(t\in [0,T]\) . Let \(U_t=\mathrm {d}\mathsf {P}_t/\mathrm {d}\Pi _n\) , \(G_t^{\pm }=\mathrm {d}\mathsf {J}^{\pm }/\mathrm {d}\vartheta _{\Pi _n}^{\pm }\) . Note that by the absolutely continuity of \(\mathsf {P}_t\)  with respect to \(d_{TV,w}\) , the map \(t\mapsto U_t\)  is absolutely continuous in \(L^1(\ell )\) . Moreover, for every \(F\in \mathcal {B}_c(\Gamma )\)  the continuity equation reads as
\(\int _{\Gamma } F (U_t-U_s) \,\mathrm {d}\Pi _n &= \int _s^t \int _{\Gamma \times \mathcal {T}} (F(\nu +\genfrac{}{}{}2{1}{n}\delta _x)-F(\nu )) G_r^+(\nu ,x) \, \mathrm {d}\vartheta _{\Pi _n}^+ \, \mathrm {d}r\\&\hspace{20.0pt} + \int _s^t \int _{\Gamma \times \mathcal {T}} (F(\nu -\genfrac{}{}{}2{1}{n}\delta _x)-F(\nu )) G_r^-(\nu ,x) \, \mathrm {d}\vartheta _{\Pi _n}^- \, \mathrm {d}r.\) 

But due to Lemma REF , the integrands can be rewritten as follows
\(\int _{\Gamma \times \mathcal {T}} F(\nu \pm \genfrac{}{}{}2{1}{n}\delta _x) G_r^{\pm }(\nu ,x) \, \mathrm {d}\vartheta _{\Pi _n}^{\pm }&= \int _{\Gamma \times \mathcal {T}} F(\nu ) G_r^{\pm }(\nu \mp \genfrac{}{}{}2{1}{n}\delta _x,x) \, \mathrm {d}(\mathsf {T}^{n,\pm }_{\#} \vartheta _{\Pi _n}^{\pm })\\&=\int _{\Gamma \times \mathcal {T}} F(\nu ) G_r^{\pm }(\nu \mp \genfrac{}{}{}2{1}{n}\delta _x,x) \, \mathrm {d}\vartheta _{\Pi _n}^\mp ,\) 

and therefore
\(\int _{\Gamma } F (U_t-U_s) \,\mathrm {d}\Pi _n &= \int _s^t \int _{\Gamma \times \mathcal {T}} F(\nu )(G_r^-(\nu +\genfrac{}{}{}2{1}{n}\delta _x,x)-G_r^+(\nu ,x)) \kappa ^+_{\nu }(\mathrm {d}x)\, \mathrm {d}\Pi _n(\mathrm {d}\nu )\, \, \mathrm {d}r\\&\hspace{20.0pt} +\int _s^t \int _{\Gamma \times \mathcal {T}} F(\nu )(G_r^+(\nu -\genfrac{}{}{}2{1}{n}\delta _x,x)-G_r^-(\nu ,x)) \, \kappa ^-_{\nu }(\mathrm {d}x)\, \mathrm {d}\Pi _n(\mathrm {d}\nu ) \, \mathrm {d}r,\) 

which is the weak formulation of (REF ). Putting in the pre-factors \((1+\nu (\mathcal {T})^2)^{-1}\)  to state the expression in terms of the finite measures \(\ell \)  and \(\Sigma \) , and noting that due to time-regularity \((1+\nu (\mathcal {T})^2)^{-1} \mathsf {P}_t\)  is TV-regular, we can proceed as in Corollary 4.14 of [1]} and conclude the proof after redefining \(U,G^{\pm }\)  on negligible sets.
